Hyperbolic jigsaws and families of pseudomodular groups, I

Abstract

We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.

Dilogarithm identities after Bridgeman

“…Remark : This identity can be interpreted as arising from the convergents of a non-standard continued fraction expansion arising from a non-arithmetic lattice Γ (1, 1/3, 3), see [7], section 7 for the definition of Γ (1, 1/3, 3) and more details. A more general version is given below.…”

Hyperbolic jigsaws and families of pseudomodular groups II

“…These are non-arithmetic surfaces/lattices which share the same cusp set as the modular surface H 2 /PSL(2, Z), that is, the cusp set is all of Q ∪ {∞}. In [5], we introduced a construction of ideal polygons (hyperbolic jigsaws) by glueing ideal triangles with marked points on the sides (jigsaw tiles) along matching sides. Such an ideal polygon (jigsaw) J is the fundamental domain for the subgroup Γ J of PSL(2, R) generated by the π-rotations about the marked points on the sides of J.…”

“…We also answer a question raised in [4] concerning the recursive formula for the tessellation of H 2 arising from punctured torus groups. As far as possible, we will adopt the notation of [5] to avoid confusion. We will recall these as necessary in the subsequent sections.…”

“…• The cases n = 2, 3 were considered in [5]. When n = 3, we were able to describe exactly which S(1, 3) jigsaw groups were arithmetic.…”

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In our previous paper [5], we introduced a hyperbolic jigsaw construction and constructed infinitely many non-commensurable, non-uniform, non-arithmetic lattices of PSL(2, R) with cusp set Q ∪ {∞} (called pseudomodular groups by Long and Reid in [4]), thus answering a question posed by Long and Reid. In this paper, we continue with our study of these jigsaw groups exploring questions of arithmeticity, pseudomodularity, and also related pseudo-euclidean and continued fraction algorithms arising from these groups.

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